Synthetic aperture radar (SAR) is a valuable imaging technique with wide-ranging applications in reconnaissance and remote sensing. It allows high resolution imaging at great distances in all weather conditions, day or night, by transmitting coherent broadband microwave radio signals from an aircraft or spacecraft platform, receiving the signals reflected from the terrain of interest, storing and processing the returns to synthesize a large aperture (allowing high resolution), and then focusing the data to form an image of the terrain's response to these radio signals. Because a SAR provides it's own illumination (microwaves) that penetrates virtually all weather disturbances, imagery can be obtained in almost all conditions.
In particular, SAR interferometry allows measurement of terrain height by examining and comparing the phase of two SAR images. These height measurements can be more accurate than the resolution of the SAR itself. SAR-derived digital elevation maps have been found to be accurate within 15 cm rms of very accurate laser theodolite and differential GPS ground surveys of the same terrain.
The fundamental key to the success of SAR imaging is that the signals transmitted and received are coherent; i.e., the transmitted signals are generated from a stable local oscillator in the radar and can be referenced in time and space to a common point. The received signals return after a precisely measurable time delay (depending in the distance to the terrain being illuminated) and they have a precisely measurable phase (or starting point on the radio wave) relative to the local oscillator reference. Coherence provides the means for synthesizing the effect of a large aperture (or antenna) which then permits much higher resolution imaging than could be provided with the physically small antenna carried on board the aircraft or spacecraft.
An important by-product of this coherent imaging (if the data are processed correctly) is that phase information is preserved in the SAR imagery that is available and can be exploited. Optical imagery, on the other hand, cannot be exploited in the same way as SAR imagery because the sun is an incoherent source of illumination and phase is not preserved.
The primary difference between optical and SAR imagery is that optical imagery shows the terrain reflectivity response to visible light whereas radar imagery shows the terrain response to radio waves.
Microwave imaging techniques, in particular SAR, can exploit the phenomena of interference to provide previously unavailable information with wide-ranging applicability. For example, SAR image collections can be processed in interferometric pairs to yield digital terrain-elevation maps accurate in relative elevation to a few centimeters. The spatial resolution of the terrain map is governed by the spatial resolution of the SAR images, typically a few feet to tens of feet. Elevation resolution is governed by the antenna separation (baseline), radar wavelength, and imaging geometry.
No other previously developed terrain elevation mapping technology (such as optical stereoscopy) can yield maps with such precision, especially considering the day/night, all weather capability and long standoff distances possible with SAR's. In addition, multiple-pass SAR image collections, with precise imaging repeat geometry but significant time lapse between collections, can be processed to detect extremely subtle sub-wavelength surface disturbances, ground motion, or other environmental changes.
Once such system is disclosed in U.S. Pat. No. 4,975,704, Dec. 4, 1990 of Gabriel et al. The process for making a SAR interferogram is set forth in Columns 2 and 3 of the patent and is incorporated herein by reference.
Phase unwrapping problems arise in scientific applications when a physical quantity (i.e., surface deformation, time delay, terrain elevation, etc.) is transduced or related to the phase of a complex signal (i.e., phase is the argument of a complex number; for example, the arctangent of the ratio of the imaginary to real parts). These signals are obtained, for example, during optical interferometry experiments, microwave interferometry (i.e., radio telescope aperture synthesis, SAR, coherent sonar beam-forming, and other applications). Because the phase is usually related in a nonlinear way to the signals being measured (i.e., arctangent computations, etc.), only the principal value of the phase can be readily obtained; that is, the computed phase lies between .+-..pi. radians. Such principal values are denoted as wrapped values. Additionally, since the phase is usually related to some physical quantity not having abrupt jumps in continuity, the wrapped phase values do not represent a useful measurement. It is necessary to remove phase discontinuities from the principal values in some logical way in order to obtain a more useful quantity; this process is known as phase unwrapping.
It is always possible to consistently unwrap samples of wrapped phases if they are considered a one-dimensional signal; i.e., a signal whose values depend on only one independent variable such as time (t) or position (x). One simply begins at the first wrapped phase sample and adds or subtracts a multiple of 2.pi. radians to the next sample so that the absolute value of the phase difference between the second and first sample is less than .pi. radians. This procedure is generally referred to as a linear path following scheme and it continues for successive samples until the entire one-dimensional array is unwrapped. No matter how it is applied to a one-dimensional signal, it will yield the same result (within an arbitrary constant offset that is usually of no consequence). In other words, the phase difference between points A and B in the now unwrapped signal will be the same, no matter how it is unwrapped.
For two or more dimensional signals, the problem is much more complicated. In the one-dimensional case just mentioned, there was only one possible unwrapping path; namely, the linear path that traverses the linear array. In the two-dimensional case, there are very many possible unwrapping paths between arbitrarily selected points A and B. If the wrapped phase values are noisy (as will be the case for any real measurement), the unwrapped values can depend on the unwrapping path taken. In other words, the phase difference between the unwrapped values at points A and B will depend on the path taken between those points. In general, it will not be possible to obtain a consistent solution with a path following scheme for multidimensional phase unwrapping. Prior art that utilizes path following techniques, such as the Gabriel et al. patent, is full of heuristic and ad hoc patches attempting to solve or mitigate the resulting inconsistencies. No robust mathematical formalism has been applied to path following methods nor are those methods likely to yield acceptable solutions to the variety of problems encountered in practice.
Two-dimensional phase unwrapping methods that did not rely on path following had their roots in adaptive optics. It was later found that the two dimensional phase unwrapping problem could be stated in a formal least-squares sense, resulting in a Poisson equation with Neumann boundary conditions and amenable to solution on rectangular grids by a host of robust numerical methods (D. Ghiglia et al., "Direct phase estimation from phase differences using fast elliptic partial differential equation solvers,": Opt. Lett. 14, 1107-09, 1989). However, all these Poisson methods solved the normal (unweighted) least-squares problem; that is, all wrapped phase values were treated with equal weight. Obviously, some wrapped phase values would be more reliable than others (due to noise, etc.) depending on the physical experiment that led to the measurements in the first place. If it were possible to assign a weight to the measured values and efficiently solve the weighted least-squares phase unwrapping problem, this would provide a significant advance to the art and completely obviate the need for any path following method.